#### Vol. 6, No. 1, 2013

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Localisation and compactness properties of the Navier–Stokes global regularity problem

### Terence Tao

Vol. 6 (2013), No. 1, 25–107
##### Abstract

In this paper we establish a number of implications between various qualitative and quantitative versions of the global regularity problem for the Navier–Stokes equations in the periodic, smooth finite energy, smooth ${H}^{1}$, Schwartz, and mild ${H}^{1}$ categories, and with or without a forcing term. In particular, we show that if one has global well-posedness in ${H}^{1}$ for the periodic Navier–Stokes problem with a forcing term, then one can obtain global regularity both for periodic and for Schwartz initial data (thus yielding a positive answer to both official formulations of the problem for the Clay Millennium Prize), and can also obtain global almost smooth solutions from smooth ${H}^{1}$ data or smooth finite energy data, although we show in this category that fully smooth solutions are not always possible. Our main new tools are localised energy and enstrophy estimates to the Navier–Stokes equation that are applicable for large data or long times, and which may be of independent interest.

##### Keywords
Navier–Stokes equation, global regularity
##### Mathematical Subject Classification 2010
Primary: 35Q30, 76D05, 76N10
##### Milestones
Revised: 7 August 2011
Accepted: 23 May 2012
Published: 1 June 2013
##### Authors
 Terence Tao Department of Mathematics University of California, Los Angeles Los Angeles, CA 90095-1555 United States